Environment random interaction of rime optimization with Nelder-Mead simplex for parameter estimation of photovoltaic models

As countries attach importance to environmental protection, clean energy has become a hot topic. Among them, solar energy, as one of the efficient and easily accessible clean energy sources, has received widespread attention. An essential component in converting solar energy into electricity are solar cells. However, a major optimization difficulty remains in precisely and effectively calculating the parameters of photovoltaic (PV) models. In this regard, this study introduces an improved rime optimization algorithm (RIME), namely ERINMRIME, which integrates the Nelder-Mead simplex (NMs) with the environment random interaction (ERI) strategy. In the later phases of ERINMRIME, the ERI strategy serves as a complementary mechanism for augmenting the solution space exploration ability of the agent. By facilitating external interactions, this method improves the algorithm’s efficacy in conducting a global search by keeping it from becoming stuck in local optima. Moreover, by incorporating NMs, ERINMRIME enhances its ability to do local searches, leading to improved space exploration. To evaluate ERINMRIME's optimization performance on PV models, this study conducted experiments on four different models: the single diode model (SDM), the double diode model (DDM), the three-diode model (TDM), and the photovoltaic (PV) module model. The experimental results show that ERINMRIME reduces root mean square error for SDM, DDM, TDM, and PV module models by 46.23%, 59.32%, 61.49%, and 23.95%, respectively, compared with the original RIME. Furthermore, this study compared ERINMRIME with nine improved classical algorithms. The results show that ERINMRIME is a remarkable competitor. Ultimately, this study evaluated the performance of ERINMRIME across three distinct commercial PV models, while considering varying irradiation and temperature conditions. The performance of ERINMRIME is superior to existing similar algorithms in different irradiation and temperature conditions. Therefore, ERINMRIME is an algorithm with great potential in identifying and recognizing unknown parameters of PV models.

incorporates the environment random interaction (ERI) strategy and the NMs mechanism to enhance the accuracy of evaluating PV models.
Since the rime particles (the individuals within the population) generated by RIME do not exist stably in the environment.The particles constantly change with the environment.The original RIME lacks the interaction of particles with the environment in the later iteration phase.In other words, these particles among RIME are not changed due to the changes in wind speed, temperature, and other external conditions in the living environment at the later stage of the algorithm.This study introduces an ERI strategy to address this issue and enhance the interaction capability between rime particles and the environment.Integrating the ERI strategy into ERINM-RIME significantly broadens the algorithm's global search space, reducing the possibility of being stuck in local optima.Including the NMs mechanism in ERINMRIME significantly enhances the algorithm's local search capability, enabling it to efficiently optimize inferior solutions and improve the overall quality of solutions.The mechanism thoroughly explores the valuable solution space, improving performance in solving complex optimization problems.Fusing the two mechanisms in ERINMRIME ultimately results in a balanced approach, achieving local and global equilibrium states.To confirm that ERINMRIME is a reliable tool for estimating PV models unknown parameters, this research conducted a comprehensive comparison with other widely used advanced algorithms for PV optimization problems.The extensive experimental results demonstrate that ERINMRIME exhibits distinctive and dominant competitiveness in addressing PV optimization problems, underscoring its superiority and efficacy in this particular domain.
Generally speaking, the following are the study's contributions: • To achieve more efficient and accurate extraction of unknown parameters in PV models, this study proposed an enhanced version of RIME, referred to as ERINMRIME.• This study introduced an ERI strategy to further interaction between rime particles and the environment in the later iteration stage.This tactic successfully reduced the algorithm's propensity to become trapped in local optima in local optima.• This study introduced the NMs mechanism, which enhanced the quality of the solutions and improved the algorithm's capability for local exploitation.• ERINMRIME was subjected to rigorous evaluation against previous advanced algorithms in PV models, affirming its reliability and value in accurately assessing PV problems.• The practical efficacy of ERINMRIME in parameter extraction from commercial PV models was assessed through testing in diverse and complex environments.
The remaining organizational structure of this paper is as follows: Section "Problem formulations" introduces the relevant PV models.Section "The proposed algorithm" comprehensively describes the proposed algorithm, detailing its key components and mechanisms.Section "Experiments and analysis for benchmark functions" verifies the effectiveness of ERINMRIME by practical experiments on IEEE CEC benchmark functions.Section "Experimental and analysis for PV models" verifies the effectiveness of ERINMRIME by practical experiments on PV models.Section "Discussion on the results" evaluates and summarizes the experimental results.Section "Conclusions and future directions" examines the potential applications of this research and provides an overview of the entire work.

Problem formulations
In the quest for improved accuracy and efficiency in assessing the I-V characteristics of PV models, researchers have dedicated their efforts to developing various models for PV systems.Within the domain of PV modeling, numerous models have been put forward.This study presents three general PV models, namely SDM, DDM, and TDM, which can be applied to all types of PV generators, including both individual cells and modules.Additionally, this section introduces the PV component models and the objective functions involved in this research.

General model
SDM When evaluating energy cases, the utilization of a precise and effective model is principal.To this end, SDM was developed to depict the PV landscape of solar energy accurately.The equivalent circuit diagram of the SDM is illustrated in Fig. 1.The model encompasses a current source operating in parallel with a diode, a shunt resistor representing the leakage current, and a series resistor accounting for the losses associated with load current 66 .The output current is calculated by Eq. (1).
where I L represents the output current of the SDM.I ph is the current generated by the light.I d represents the current through the diode, and I sh represents the current through the shunt resistor.I d is calculated as shown in Eq. (2).
where I sd represents the reverse saturation current.q is the elementary charge ( 1.60217646 × 10 −19 C ). V L is the output voltage.R s represents the series resistance.n is the ideal factor of the diode.k is the Boltzmann constant ( 1.3806503 × 10 23 J/K ), and T is the temperature in Kelvin.The current I sh through the shunt resistor can be obtained by Eq. ( 3).

DDM
Due to the drawbacks of the SDM, such as neglecting the temperature effect, lacking accurate nonlinear description, and limited parameterization flexibility, this study introduces the DDM as the benchmark model to enhance the universality and comparability of research results, facilitating easier comparison and evaluation of different algorithms' performance.
The DDM incorporates a diode and a photogenerated current source in series to simulate the compound current 67 .The equivalent circuit diagram of the DDM is depicted in Fig. 2. The final current generated by DDM is shown in Eq. (5).
where n 1 and n 2 represent the ideal coefficient of diffusion and composite diode.I sd1 and I sd2 represent the diode diffusion and saturation current, and other parameters are the same as the above formula, respectively.TDM While the DDM offers improved accuracy, it is still deemed inadequate in accurately resolving large-scale and intricate engineering challenges.This study introduces the more precise TDM as a benchmark model to further improve the accuracy and comparability of research results.By incorporating the TDM, algorithm performance and robustness can be better evaluated, especially under complex operating conditions.This contributes to advancing research and development in photovoltaic systems and facilitates comparisons and evaluations among different algorithms.
The TDM incorporates an additional diode based on the DDM 68 .The equivalent circuit diagram of the TDM is illustrated in Fig. 3.The ultimate output current of the TDM can be obtained by Eq. ( 7).
(3)  where I sd3 and n 3 represent the current flowing through the third diode and its ideal factor.

PV module model
The PV module model typically consists of multiple models interconnected in a series or parallel arrangement 69 .
The equivalent circuit of the PV module model is presented in Fig. 4. The output current of the PV module model, based on the SDM and DDM, can be mathematically expressed by Eq. ( 8) and Eq. ( 9), respectively.
where N p is the number of parallel solar cells.N s is the number of series solar cells.

Objective function
To improve the fitting of the data of the I L and V L of PV models, this study adopts the Root Mean Square Error (RMSE) as the optimization objective to minimize the discrepancies between the measured data and the real values.A smaller RMSE indicates a better fit of the data.The specific formula for RMSE is as follows: (7)  where I L and V L are the output current and voltage of the model, respectively.N represents the amount of meas- ured current data.X represents the solution vector containing unknown parameters.Equation (11)-Eq.( 13) represent the RMSE functions ( f ) for SDM, DDM, and TDM, respectively.

Ethical Statement
The manuscript has not been submitted to more than one journal for simultaneous consideration and has not been published elsewhere in any form or language.

The proposed algorithm
This section shall elucidate the underlying principle of the RIME, along with the incorporated environment random interaction (ERI) strategy, the NMs mechanism, and the upgraded algorithm, referred to as ERINMRIME.

Soft-rime search strategy
A highly stochastic process characterizes the growth of soft rime.Rime particles exhibit free movement across the surface of the attached objects, with a tendency to grow gradually in a specific direction.Therefore, a novel soft rime search strategy has been proposed to simulate the inherent randomness and extensive coverage of soft rime accurately.It enables RIME to perform global searches more comprehensively.The specific updating formula of rime is described as follows: where R new ij is the new position of the rime particles updated.I represents the ith rime particle.J represents the jth dimension.Ub i,j and Lb ij represent the upper and lower bounds of the ith rime particle in the jth dimension, respectively.R best,j is the jth dimension of the optimal rime particles.r 1 , r 2 and h are three random numbers in the range (-1, 1), independently.θ , β and E change with the number of iterations.Their specific definitions are as follows: where t is the number of current iterations.W sets a constant, w = 5 .T represents the maximum number of iterations.

Hard-rime puncture mechanism
In contrast to soft rime growth, the growth of hard rime is relatively straightforward and more regular.Each rime agent passes through easily due to the same direction of growth, which leads to the phenomenon known as rime puncture.RIME proposes the mechanism of hard rime puncture to accurately model the growth of hard rime when rime particles condense into this state.This mechanism enhances the local exploration ability of the RIME, by expanding the solution space and enabling more effective sampling of the local search area.
The specific updating formula of rime particles is shown below. (10) www.nature.com/scientificreports/where R new ij is the newly updated position of the rime particles.r 3 is a random number in the range (-1, 1).R best,j is the jth dimension of the optimal rime particles.F normr (S i ) is the normalized value of the current particle fit- ness value.

The positive greedy selection mechanism (PGSM)
The PGSM is an improvement on the original greedy selection mechanism.The updated fitness value F(R new i ) of the particle is compared with that of the particle before the update F(R i ) .If the updated fitness value is better than that before the update, F(R i ) will be replaced and the solution of two particles will be replaced.Eventually, the fitness value of the optimal solution F(R best ) and the corresponding solution are updated.By incorporating the PGSM, the algorithm ensures that the population consistently progresses toward a globally optimal solution at every iteration.The schematic diagram of RIME is shown in Fig. 5.

ERI strategy
This section introduced the ERI strategy, which is a global exploratory approach.Although RIME takes into account the different formation methods and morphologies of the two types of rime, the newly formed rime will change its form due to further changes in the environment.Rime particles are susceptible to changes in wind speed or temperature, which can cause them to undergo transformations and change into other forms of rime particles in the air.The RIME lacks further interaction with the environment after the formation of rime particles, which leads to the algorithm become stuck in local optima.To address this issue, this study introduces the ERI strategy 70 to achieve further interaction with the environment.In this strategy, two random solutions are generated to control the exploration area of the algorithm.This strategy enables a thorough exploration of the surrounding regions of each individual within a certain distance, facilitating a global search effect throughout the entire algorithm.Therefore, the newly formed rime particles will undergo spontaneous modifications in response to environmental fluctuations.This ensures that the algorithm can quickly break free from local optima in a timely manner and explore the solution space more comprehensively.The following detailed description of the policy is a a full description of the policy.Fig. 6 displays the ERI strategy's schematic diagram.
where R new 1 and R new 2 are two particles randomly selected from the newly created rime particles.diff represents the midpoint between the two random solutions.It controls the distance by which each individual further explores its surrounding area.The assignment of parameters diff aids in guiding the exploration process, enabling the algorithm to effectively explore the solution space with a balanced approach.R new i is the ith solution of the new solution.R en 1 and R en 2 represent the solution generated by two randomly selected particles after environmental interaction.
(18) Comparing the fitness values of solutions generated through environmental interaction, the ERINMRIME selects the relatively optimal solution and eventually compares it with F(R new i ) .If the temp is better, the algorithm updates R new i .This ensures that the solution is moving in the direction of the global optimal solution.

NMs
The NMs mechanism 71 is a well-established approach for solving optimization problems.Due to its remarkable local search capability, it has been extensively employed in the realm of PV model parameter identification.As a consequence, this study uses the NMs mechanism to boost the local exploitation capability of ERINMRIME.A simplex can be visualized as a polyhedron with n + 1 vertices, which forms the convex hull of n dimensions.The NMs mechanism utilizes this geometric concept to improve the current position of the solution by performing a sequence of operations such as reflection, expansion, and contraction, which ultimately leads to the formation of a new simplex with improved properties.By repeating this process iteratively, the NMs mechanism can efficiently search the solution space and converge toward the ideal solution.It continuously refines the solutions, guiding the algorithm to estimate the parameters in the PV models more accurately.The following are the actual steps of the NMs mechanism: Fig. 7 displays the schematic diagram of NMs.
1. Choice.Each individual is sorted according to fitness value and all individuals are numbered from 1 to n + 1.
As shown in Eq. (26).The mechanism selects the optimal two individuals and averages them to get the position of the simplex centroid.The specific calculation formula is shown in Eq. ( 27).
(   where α is the reflection coefficient.If F(R 1 ) < F(R r ) < F(R n ) , the mechanism performs R n+1 = R r .In other words, if the newly generated reflection position is between the simplex optimal value and the subdifferential solution, it means that the worst solution has been promoted.3. Extension.If F(R r ) > F(R 1 ) , the exploration area is further expanded by combining the center of mass R c and the reflection position R r .In other words, if F(R r ) > F(R 1 ) , it indicates that the best way to find the global optimal solution is to search in the direction of reflection.
where R e is the expanded position.β is the expansion coefficient.If F(R e ) < F(R r ) , the mechanism performs R n+1 = R e .Otherwise, the mechanism performs R n+1 = R r .In other words, if the expanded position is the position of the centroid, it means that the expansion is valuable and the worst solution is updated.Otherwise, the newly generated position returns to the origin position.The worst solution is equal to the position of the centroid.4.
where γ is the shrinkage coefficient.If F(R oc ) < F(R r ) , the mechanism makes R n+1 = R oc .Similar to expan- sion, if the contracted position is better than the centroid position, the worst solution is updated to the contracted position.Otherwise, the mechanism continues to execute step 5 to try internal shrinkage.5. Contraction.If F(R r ) > F(R n+1 ) is true, the mechanism performs internal shrinkage according to Eq. (31).

The proposed RIME-based algorithm
To address the existing limitations of the RIME, such as low convergence accuracy and the propensity to become stuck in local optima, this study has incorporated two novel strategies, namely the ERI strategy and the NMs mechanism.On the one hand, the ERI strategy leverages two random solutions to regulate the algorithm's ability to explore the world, safeguarding it against becoming stuck in local optima and enhancing its capacity for global search.On the other hand, the NMs mechanism employs a scaling transformation centered around the current best solution to focus on the local region and obtain higher-precision solutions, thereby bolstering the algorithm's capability for local exploitation.The amalgamation of these two mechanisms fosters a higher degree of equilibrium in the algorithm, substantially enhancing its convergence accuracy.ERINMRIME mainly comprises the following procedural steps: Step 1: Initialize the position of the rime particles using Eq.(32).
where rand is a random number in the range of (0, 1).ub and lb are the upper and lower boundaries respectively.N is the population size.R i represents the generated ith rime particle.
Step 2: Fitness values are calculated for randomly generated rime populations to screen out the optimal particles.
Step 3: At the commencement of each iteration, the algorithm executes the soft rime strategy, following the principles outlined in Eq. ( 14)- (17), which facilitates the updating of the rime population and helps the algorithm perform a global search.
Step 4: The algorithm determines whether to employ the hard-rime puncture mechanism based on the conditions specified in Eq. ( 18).This mechanism aims to conduct further exploration within the local solution space, where a potential globally optimal solution may exist.
Step 5: According to Eq. ( 19)-( 20), the algorithm implements the PGSM to select the newly updated rime population to ensure that the population renewal is always moving in the right direction.The value of the optimal rime particle is updated at the same time.
Step 6: According to Eq. ( 21)-( 25), the ERI strategy of the environment is implemented.By means of the haphazard engagement with the surroundings, the rime particles in the solution space can be transformed randomly, which makes the algorithm explore the solution space more fully.
Step 7: According to Eq. ( 26)-Eq.( 31), the algorithm effectively executes the NMs mechanism.This mechanism guides the algorithm towards the global optimal solution, further enhancing solution quality through simplex transformations.Moreover, it promotes the local development capability of ERINMRIME, ultimately achieving a deeper level of balance in the algorithm's performance.
Step 8: Iterate through the aforementioned process until the specified stopping condition is met.
To provide a clear illustration of the procedural flow of ERINMRIME, the algorithmic flowchart is depicted in Fig. 8, while the detailed pseudo-code of the algorithm is presented in Algorithm A1 (Online Appendix).

Complexity analysis
Algorithm complexity analysis is a crucial step in evaluating and optimizing algorithm performance.It quantitatively assesses algorithm efficiency, resource requirements, and scalability, helping us make informed decisions in algorithm design and selection 72 .
In this study, the complexity of the proposed ERINMRIME was analyzed using the method proposed by Ni et al. 73

Experiments and analysis for benchmark functions
To validate the optimization performance of ERINMRIME, this study conducted experiments on the benchmark functions of IEEE CEC 2017 and IEEE CEC 2020.ERINMRIME was compared with other improved algorithms in these experiments.

IEEE CEC 2017 and IEEE CEC 2020
The experiments were carried out independently on IEEE CEC 2017 and IEEE CEC 2020 benchmark functions in order to guarantee impartiality and thoroughness.
IEEE CEC 2017: It consists of benchmark functions primarily categorized into three types.F 1 − F 3 represents unimodal functions.F 4 − F 10 represents multimodal functions.F 11 − F 20 represents hybrid functions.F 21 − F 30 represents composition functions (a combination of functions from different test problems) 74 .Table 1 comprehensively describes the 30 benchmark functions included in IEEE CEC 2017.
All benchmark functions are formulated as unconstrained minimization problems.The mathematical model is represented as follows: where f (X) is the objective function.X is the decision variable vector.x j represents the decision variable within the range lb j ≤ x j ≤ ub j , j = (1, . . ., D) .D represents the dimension.S is the decision space, which ranges from -100 to 100.
IEEE CEC 2020: This collection contains 10 benchmark functions.Table 2 provides a comprehensive description of the 10 benchmark functions included in IEEE CEC 2020.

Measure metrics
To ensure a more reliable assessment of algorithm performance, this study employed various metrics such as average value (Avg) and standard deviation (Std).Convergence curves were also compared to analyze the experimental results.The best results among the tested experiments were highlighted in bold for easy identification.Moreover, non-parametric statistical tests were employed to ascertain the statistical significance of the enhanced algorithms, specifically the Wilcoxon Signed Rank Test (WSRT) 75 and the Friedman Test (FT) 76 .The p-value was set to 0.05.A p-value of less than 0.05 suggests a significant difference between the two algorithms 77 .This significance level ensures that there is less than a five percent probability for test 78 .

Parameter settings
Experiments were conducted using MATLAB R2018b software on a Windows 11 operating system to ensure fairness in the experiments.The hardware specifications included an Intel i7-12700H CPU and 16 GB of RAM.
In addition, the effectiveness of the algorithm performance evaluation is based on the fairness and rationality of the experimental setting.Thus, the parameters for the experiments were kept consistent.The population size (N) was set to 30, the issue dimension (D) was 30, and the maximum iteration count (MaxIt) was set to 300,000.This implies that the search agent evaluations were performed a total of 300,000 times.The upper boundary (ub) and lower boundary (lb) of the search space were established at 100 and -100, correspondingly.Additionally, each function involved in the experiments was also independently tested 30 times.
The specific parameters of ERINMRIME proposed in this paper are directly adopted from the original RIME values.No new parameters were introduced in the ERI strategy and NMs mechanism.

Experimental results on IEEE CEC 2017
To assess ERINMRIME's optimization performance, this section compared it to other improved algorithms using the IEEE CEC 2017 benchmark functions of IEEE CEC 2017.The comparison algorithms including m_SCA, SCADE, ASCA_PSO, OBSCA, HGWO, RCBA, and CBA.The parameter settings for each algorithm are provided in Table 3.
Table 4 presents the average values and standard deviations of 8 algorithms.ERINMRIME performs poorly on composite functions in the table but achieves the best or near-best average values on the remaining benchmark functions.Considering the comparison results of ERINMRIME with the other seven algorithms in terms of WSRT and overall ranking in Table 5, ERINMRIME ranks first overall, indicating its significant optimization capability.Generally speaking, WSRT whether the algorithms have statistical significance in their test functions.In Table 5, symbols "+/=/−" indicate whether ERINMRIME is superior to, equal to, or lower than other algorithms, respectively.ERINMRIME obtains a WSRT value of 3.43 and significantly outperforms other improved algorithms on over half of the benchmark functions out of the 30.This indicates that ERINMRIME significantly differs from the other seven algorithms on most functions.
Figure 9 depicts the bar chart of the FT rankings of the 8 algorithms.The p-value of ERINMRIME is 3.44, which is the lowest compared to other improved algorithms.FT can assess the stability of the algorithm's optimization performance.Therefore, this also indicates that ERINMRIME exhibits a certain level of stability.Figure 10 shows the convergence curves of ERINMRIME and the other 7 algorithms.From the red line in the graph, ERINMRIME exhibits the maximum convergence accuracy, as evidenced by its persistent placement at the highest convergence accuracy.Furthermore, the steepness of the red curve's descent in the graph illustrates that ERINMRIME has the fastest convergence speed, allowing it to converge toward the optimal solution rapidly.To sum up, ERINMRIME can be proven to be an effective optimization algorithm, possessing stable optimization performance and significant competitiveness.

Experimental results on IEEE CEC 2020
To further verify the algorithm's optimization efficiency in this section, ERINMRIME is compared with other improved algorithms on the IEEE CEC 2020 benchmark functions of IEEE CEC 2020.The average values and standard deviations for each of the eight algorithms are shown in Table 6.Within the table, ERINMRIME achieves the optimal average value on half of the 10 benchmark functions in IEEE CEC 2020.This demonstrates the excellent optimization capability of this algorithm.Table 7 presents the WSRT values and overall rankings of

Experimental and analysis for PV models
To assess the effectiveness and reliability of ERINMRIME for parameter identification of PV models, a series of experiments are conducted on ERINMRIME, comparing it with other enhanced algorithms in the context of SDM, DDM, TDM, and the PV module model.Subsequently, to provide a more robust demonstration of ERINMRIME's efficacy in practical scenarios, this study conducts further testing of the algorithm using three commercial PV models.In this study, the experiments were all implemented on MATLAB R2018b software.The operating system was Windows 10.The processor was Intel i7-6700HQ (3.40 GHz), and the memory was 16 GB.The basic parameters of the experiment were the same, the population size (N) was 30, the dimension (dim) of the problem was 30, the maximum number of iterations (MaxIt) was set to 20,000, and each involved function was independently tested for 30 times.This fair experimental setting reduces the impact of test environment deviation on the final experimental results.In the experimental setup, Table 8 presents the defined upper and lower bounds for each parameter.The RMSE is adopted as the evaluation metric to quantify the disparity between predicted and actual values.Moreover, the symbol " + " indicates that ERINMRIME outperforms the compared algorithms, while "=" denotes equivalent performance.Throughout the experimental process, the Absolute Error (IAE) and Relative Error (RE) are calculated in order to evalute and easure the differences between the experimental data and the estimated data.The expressions for I IAE and I RE , P IAE and P RE can be written as follows: where I measure represents the actual current value.I simulate represents the estimated current value.P measure indicates the actual power value.P simulate and indicates the estimated power value.
Friedman test

Results on the SDM case
This study compares ERINMRIME with other advanced algorithms, including IJAYA 86 , GOTLBO 87 , OLGBO 88 , TLBOBSA 64 , GOFPANM 89 , MLBSA 90 , EHHO 91 , Dwarf Mongoose Optimizer (DMO) 92 , Artificial Hummingbird Algorithm (AHA) 93 , Social Network Search (SNS) 94 , Mantis Search Algorithm (MSA) 95 , BSA 96 , RIME 44 .Particular experimental outcomes are displayed in Table 9.The RMSE by ERINMRIME significantly decreases compared to the RIME.The phenomenon indicates that the accuracy of ERINMRIME has been markedly promoted.Furthermore, ERINMRIME achieved an RMSE value of 9.86022E-04, comparable to the most recent advancements in algorithmic improvements, including OLGBO, TLBOBSA, GOFPANM, and MLBSA.This also reflects that ERINMRIME is worthy of deep exploration in parameter extraction of SDM, which has excellent development potential.To observe the benefits and drawbacks of the comparison method more intuitively, the image of ERINMRIME with its comparison algorithm on the RMSE standard is shown in Fig. 13.It is apparent that in the algorithm's first phase, the convergence speed of ERINMRIME is second only to GOFPANM.But when more assessments are made, ERINMRIME progressively approaches and eventually surpasses GOFPANM, yielding the optimal result.Meanwhile, the image clearly illustrates that the purple curve of RIME quickly falls into a local optimum, indicating limited optimization capability.However, the red curve representing ERINM-RIME successfully avoids local optima, demonstrating a substantial enhancement in the algorithm's optimization capabilities compared to the original RIME.In conclusion, ERINMRIME demonstrates outstanding performance and absolute superiority in the extraction of SDM parameters.The fitting comparison of the actual measurement data and simulated data of ERINMRIME on the SDM is shown in Fig. 14.The figure demonstrates that the estimated data produced by ERINMRIME exhibits a high degree of fitting with the actual data.Figure 14

Results on the DDM case
Likewise, ERINMRIME was also evaluated on the DDM.Table 10 shows the results of the tests.When compared to other algorithms, the table clearly shows that ERINMRIME delivers the lowest RMSE, signifying superior performance in the testing phase.Furthermore, its improvement compared to the original RIME algorithm is remarkably significant.In Fig. 17, the specific convergence images of the comparative algorithms are presented, providing a visual representation of ERINMRIME's effectiveness.The graph showcases how ERINMRIME outperforms other algorithms, achieving better convergence and optimization capabilities.The early convergence speed of ERINMRIME is notably faster.While ERINMRIME and GOFPANM exhibit similar trends during the middle and later phases of the algorithm, ERINMRIME outperforms GOFPANM in terms of achieving higher   characteristic curve, where the real measured data and simulated data of I-V and P-V from ERINMRIME on the DDM are compared.It is noteworthy that there is a remarkable level of consistency in the fitting of these two forms of data.In Fig. 19, a comparison is presented between the specific data of IAE and RE of I predicted by ERINMRIME and the corresponding actual data.Detailed data results are presented in Table A2 (Online Appendix).The data that was previously shown makes it clear there is a high degree of agreement in the estimation of unknown data.The ERINMRIME's performance in predicting and estimating unknown values is highly accurate and reliable.The error curve between power and voltage is shown in Fig. 20.To sum up, ERINMRIME has high precision in measuring unknown parameters on DDM.

on the TDM case
Similarly, ERINMRIME was applied to the TDM for testing.Table 11 displays the findings of the final comparison.ERINMRIME possesses the optimal RMSE.As depicted in Fig. 21, it is evident that the ERINMRIME convergence curve is located around the graph's bottom.The ERINMRIME has the fastest convergence speed.ERIN-MRIME has a finite amount of iterations to reach optimal values, demonstrating its strong early optimization capabilities and ability to obtain relatively accurate results quickly.In addition, ERINMRIME owns the smallest RMSE, and the converge accuracy is unquestionable.Therefore, ERINMTIME is an effective tool for accurately estimating unknown parameters on TDM.25.From the table, it is clear that ERINMRIME's RMSE shows a significant improvement compared to RIME.An order of magnitude reduces the RMSE from ERINMRIME.Despite having a similar RMSE to the other four common algorithms, the convergence images demonstrate that ERINMRIME achieves the highest final convergence accuracy among all the algorithms.According to the results, ERINMRIME surpasses the other algorithms in terms of optimization precision and accuracy.Meanwhile, the convergence curve of ERINMRIME has a noticeable decline trend, and the convergence speed is good.The optimal solution can be quickly found, and this optimal value is the   minimum algorithms.This demonstrates that ERINMRIME has achieved better results in terms of both convergence speed and accuracy.

Results on the PV module model cases
In addition, this study compares the data estimated by ERINMRIME with the actual data and finally gets 25 groups of specific experimental results, as displayed in Table A4 (Online Appendix).The specific fitting image of the actual and estimated data is shown in Fig. 26.Apparently, the fitting of the two is highly coincident.Correspondingly, Fig. 27 and Fig. 28 illustrate the variations in IAE and RE for ERINMRIME concerning the parameters I and P at various voltage levels.The findings of the experiment results show that ERINMRIME can accurately identify PV module model parameters.In conclusion, ERINMRIME performs remarkably well in various scenarios, making it a valuable algorithm for the in-depth exploration of PV model parameter recognition.Its effectiveness and versatility suggest that ERINMRIME is a promising choice for tackling complex PV modeling and optimization challenges.

Time comparison between ERINMRIME and other algorithms
This study compares the running times of each algorithm on different models, as shown in Table 13.ERINM-RIME exhibits comparable running times to other algorithms, but its convergence accuracy is excellent.Although GOFPANM is closest to the convergence accuracy of ERINMRIME, ERINMRIME runs about 300 times faster than GOFPANM.Based on the comprehensive examination of each model's experimental outcomes provided above, it is determined that ERINMRIME has the fastest convergence speed among them.ERINMRIME can converge quickly with appropriate precision in relatively little time, which benefits from adding an ERI strategy.It enables ERINMRIME to explore the region fully through simple operations to expand its broader exploration   of the algorithm.Simultaneously, the NMs mechanism also develops local areas in depth with low computational complexity.

Results based on the manufacturer's datasheet
ERINMRIME was used to test the efficacy of SDM and DDM's parameter estimates on three widely used commercial model components, namely single crystal SM55 97 , thin film ST40 98 , and polycrystal KC200GT 99 .The results further demonstrate the practicality reliability of ERINMRIME in estimating parameters for various PV models, solidifying its potential for real-world applications.The experiments were tested under different irradiance and temperatures, respectively.This comprehensive testing approach ensures that the algorithm can effectively handle the variations in irradiance and temperature, making it reliable for real-world applications in diverse operating conditions.Among them, the best experimental results are indicated in bold.The respective PV model manufacturer supplied each of the three models's data sets.

Results for Thin-film SM55 datasheet
The SM55 is a solar cell consisting of 36 PowerMax® single crystal series.The SM55 used in this study is produced by Shell Solar company.Please refer to the literature for specific specifications and parameters 97 .
Table A5 (Online Appendix) shows the specific parameters list of ERINMRIME under different irradiation at 25℃.To thoroughly manifest the accuracy of the ERINMRIME in estimating PV model parameters, this    29 (c, d) displays the specific current and voltage values of the ERINMRIME based on the SDM or DDM battery model on SM55 at different temperatures (25 ℃, 40 ℃, 60 ℃) under the condition of the fixed variable 1000 W/m 2 .Noticeably, the estimation of series parameters of the SM55 by ERINMRIME is consistent with the height of the actual measured data.Similarly, in Fig. 29 (a, b), when different irradiances are tested at a fixed variable of 25 ℃, it can be distinctly found that the relative transformation of irradiation almost does not influence the measurement of the algorithm.To sum up, the estimation ability of ERINMRIME in the actual PV models has been greatly affirmed.It is a valuable algorithm worthy of further exploration in the measurement of PV model parameters.The ST40 used in this study was produced by Shell Solar.Please refer to the literature for specific specifications and parameters 98 .Table A6 (Online Appendix) displays the list of specific parameters of different irradiation of ERINMRIME at 25 ℃. Figure 30 (a, b) shows the fitting images of the measured data and estimated data acquired by ERINM-RIME at ST40 consisting of SDM or DDM.ERINMRIME's accuracy is guaranteed under a variety of irradiation conditions.There will be no relatively large change in accuracy with the irradiation change.This study compared ERINMRIME and other algorithms based on the RMSE assessment criterion in order to better illustrate the estimation algorithm's accuracy.As shown in Table 16, the RMSE of ERINMRIME was always the smallest.The In addition, study examined results of PV parameters measured by ERINMRIME at the fixed irradiation condition of 1000 W/m 2 under four different temperatures to confirm the influence of temperature on the measured data.Table 17 displays the specific values.Fig. 30 (c, d) displays the fitting curve between the observed values and the corresponding specified current and voltage values.From the tables and images, it is evident that the estimated data of ERINMRIME shows a high level of fit with real data.To further validate the competitiveness of ERINMRIME, this study conducted a comparison with other classic algorithms at the four different temperatures previously indicated, confirming its superiority.Ultimately, the algorithm demonstrated prominent advantages over the other methods in accurately estimating the PV models in various temperature conditions.

Results for Thin-film KC200GT datasheet
An efficient polycrystalline silicon PV model is the KC200GT.The KC200GT used in this study is produced by KYOCERA Corporation.Please refer to the literature for specific specifications and parameters 99 .
To further verify the accuracy of measurements generated by ERINMRIME on the PV models.This study conducted deep testing on the commercial model KC200GT.Table A7 (Online Appendix) reveals the specific parameter list of ERINMRIME under different irradiations at 25 ℃. Figure 31 (a, b) shows fitted images consisting of actual measurement of PV models of SDM or DDM KC200GT corresponding to measured data of ERINMRIME.It can be distinctly seen that the error between the two kinds of data is very small, which can guarantee the corresponding relationship of accuracy.Moreover, the measurement results of ERINMRIME will not have relatively large precision changes with the changes of irradiation.To further demonstrate the improvement of the estimation data of the algorithm, this study compared ERINMRIME with other algorithms on the evaluation standard of RMSE.As shown in Table 18, the RMSE of ERINMRIME is always the smallest among them.ERINMRIME is an algorithm with improvement significance, strong competitiveness and optimization ability.
Furthermore, this study tested the algorithm results under fixed irradiation conditions 1000 W/m 2 at three different temperatures in order to thoroughly verify the impact of temperature on the recorded data.The corresponding measured values can be found in Table A10 (Online Appendix).Fig. 31 (c, d) displays the fitting curve between the observed values and the particular current and voltage values.As evident from the figure, the estimated data obtained by the algorithm exhibit a strong degree of conformity with the actual data.For proving the unique advantages of ERINMRIME and the relative competitiveness of optimization ability, ERINMRIME was compared with other commonly used traditional algorithms under the above three different temperatures.The eventual comparison outcomes are displayed in Table 19.Upon inspection of the bolded data, the ERINMRIME

Discussion on
For the of solving the problem of RIME being susceptible to getting trapped in local optima during later iterations, this study introduces the ERI strategy.The unstable form of rime particles deeply interacts with the environment, making rime particles wander more uniformly in the global space of the solution.This strategy facilitates further interaction between the newly generated rime particles and the environment, thereby enhancing the algorithm's capacity to search the solution space and stay out of local optima.In order to balance between exploration and exploitation and to explore the selected valuable local solution space in greater depth, the NMs mechanism is introduced in this study to improve the local search capability of the algorithm.NMs mechanism can optimize the optimal solution, improving its quality.Combining these two mechanisms optimizes RIME, leading to a highly effective and efficient algorithm known as ERINMRIME.The incorporation of the ERI strategy and NMs mechanism has garnered positive feedback and demonstrated exceptional performance in the parameter extraction of PV models.
To showcase ERINMRIME's efficacy on PV models, this study carried out experiments on various models.ERINMRIME was compared against other advanced algorithms, namely OLGBO, TLBOBSA, IJAYA, GOTLBO, GOFPANM, MLBSA, EHHO, BSA, and RIME, to comprehensively assess its performance and superiority in parameter extraction and optimization tasks.ERI strategy enables rime particles to further communicate with the environment, making the newly generated particles more closely integrated with the environment and climate.The incorporation of the ERI in the optimization of RIME ensures the avoidance of local optima and enhancement of the global search capability of the ERINMRIME.The ERI strategy facilitates thorough exploration of the solution space, while the NMs mechanism iteratively explores the vicinity of promising solutions, enabling ERINMRIME to develop and refine locally.The synergistic combination of these two mechanisms complements each other, resulting in a substantial improvement in the optimization capabilities of ERINMRIME, taking its performance to a higher level.The final experimental results indicate that ERINMRIME is a robust optimization algorithm.In addition, to verify the practicality of the ERINMRIME in practical PV commercial models, this study used three real datasets SM55, ST40, and KC200GT extracted from the manufacturer's dataset to test the PV models respectively.The simulated values obtained through the fitting process are compared with the corresponding real values.The model parameters estimated by ERINMRIME showed a highly fitting state with the actual data.To further confirm the optimization value of this algorithm, this study evaluated it against other traditionalal algorithms for RMSE under various temperatures and irradiation levels.ERINMRIME always obtained a lower RMSE value.In conclusion, the ERINMRIME proposed in this study has exhibited remarkable competitiveness in identifying unknown parameters in PV models.ERINMRIME is an algorithm worth exploring further for its practical value.

Conclusions and future directions
The present study introduces an enhanced version of the RIME, referred as ERINMRIME, that integrates the strategy with NMs mechanism.The incorporates a random interaction of the environment, which facilitates thorough interactions between the newly generated RIME particles and the environment.This enhances ERINMRIME's ability to search globally, enabling it to avoid local optima.The ERINMRIME's capacity for local exploitation can be enhanced by the NMs mechanism.The two mechanisms complement each other and finally raises the algorithm's overall performance.To verify the ERINMRIME's performance, this study conducted experiments on PV models.Comparing the findings to the original RIME algorithm, there are noticeable performance gains.Specifically, ERINMRIME achieved performance gains of 46.23%, 59.32%, 61.49%, and 23.95% for SDM, DDM, TDM, and the PV module model, respectively, showcasing its remarkable capabilities in optimizing and extracting parameters for various PV models.In addition, the algorithm was also validated on three commercial models.In the end, the experimental findings show that ERINMRIME performs remarkably well when it comes PV model parameter extraction.While ERINMRIME has demonstrated satisfactory performance in parameter optimization for PV models, it may not be as effective in tasks such as object tracking, energy optimization, and multi-objective optimization.In future work, other optimization problems and objectives should be considered.Additionally, addressing multiobjective problems is a hot topic, but it requires balancing and optimizing multiple conflicting objective functions to obtain a set of optimal solutions, which differs from single-objective optimization.Therefore, developing a multi-objective version of ERINMRIME will be a future direction of research.
. The complexity of ERINMRIME is mainly determined by key factors such as the maximum number of iterations (MaxIt), population size (N), and problem dimension (D).The main components of ERINMRIME include the initialization of fitness values, RIME's original composition, NMs mechanism, and ERI strategy.The complexity of initializing fitness values is O(N) .The complexity of RIME is known to be O((N + log N) * N * MaxIt) based on the original paper.The complexity of the ERI strategy is O(N * MaxIt) .The complexity of the NMs mechanism is O(D * MaxIt) .Combining these complexities, the overall complexity of ERINMRIME is O(N + MaxIt * (N * (N + log N) + N + D)). https://doi.org/10.1038/s41598-024-65292-xwww.nature.com/scientificreports/

Figure 9 .
Figure 9.The outcome of ERINMRIME's FT in comparison to seven algorithms at IEEE CEC 2017.
(a) and Fig. 14 (b) display the I and P values of ERINMRIME on the SDM with voltage change, respectively.The specific experimental current and power data are shown in

Figure 10 .
Figure 10.ERINMRIME's convergence curves in comparison to seven different algorithms at IEEE CEC 2017.

Figure 15 (
Figure 15 (a) and Fig. 15 (b) reveal the values of I IAE and I RE that vary with volt- age, respectively.The corresponding diagram about P IAE and P RE are shown in Fig. 16.Overall, ERINMRIME exhibits exceptional performance relative to other algorithms due to the perfect combination of two mechanics with RIME.The ERI strategy can help the algorithm avoid local optimization, which allows rime particles to explore new solution space further.NMs mechanism can deeply explore the search for valuable solution regions that have been screened out.The combination of the two characteristics has further boosted the algorithm.The experiments on SDM have also demonstrated the value of adding the above two mechanisms.

Figure 11 .
Figure 11.The outcome of the FT of ERINMRIME on IEEE CEC 2020 in comparison with seven methods.

Figure 14 .
Figure 14.(a) The I-V curve of SDM simulated by ERINMRIME (b) The P-V graph gained by ERINMRIME.

Figure 15 .
Figure 15.(a) The IAE value of I with voltage chart of ERINMRIME (b) The RE value of I with voltage graph of ERINMRIME.

Figure 17 .
Figure 17.The convergence curve on the DDM.

Figure 18 .
Figure 18.(a) The I-V curve of DDM simulated by ERINMRIME (b) The P-V graph gained by ERINMRIME.

Figure 19 .Figure 20 .
Figure 19.(a) The IAE value of I with voltage chart of ERINMRIME (b) The RE value of I with voltage graph of ERINMRIME.

Figure 22 .
Figure 22.(a) The I-V curve of TDM simulated by ERINMRIME (b) The P-V graph gained by ERINMRIME.

Figure 23 .Figure 24 .
Figure 23.(a) The IAE value of I with voltage chart of ERINMRIME (b) The RE value of I with voltage graph of ERINMRIME.

Figure 26 .
Figure 26.(a) The I-V curve of PV simulated by ERINMRIME (b) The P-V graph gained by ERINMRIME.

Figure 27 .
Figure 27.(a) The IAE value of I with voltage chart of ERINMRIME (b) The RE value of I with voltage graph of ERINMRIME.

Figure 28 .
Figure 28.(a) The IAE value of P with voltage diagram of ERINMRIME (b) The RE value of P with voltage graph of ERINMRIME.

Figure 29 .
Figure 29.(a, b) The I-V curves of SDM and DDM for SM55 simulated by ERINMRIME at 25 ℃ under various irradiances (c, d) The I-V graph of SDM and DDM acquired by ERINMRIME at various temperatures in the irradiance of 1000 W/m 2 .

Figure 30 .
Figure 30.(a, b) The I-V curves of SDM and DDM for ST40 simulated by ERINMRIME at 25 ℃ under varied irradiances (c, d) The I-V graph of SDM and DDM obtained by ERINMRIME at various temperatures in the irradiance of 1000 W/m 2 .

Figure 31 .
Figure 31.(a, b) The I-V curves of SDM and DDM for KC200GT simulated by ERINMRIME at 25 ℃ under diverse irradiances (c, d) The I-V graph of and gained by ERINMRIME at various temperatures in the of 1000 W/m .
The reflection position R r is constructed by combining the center of mass R c and the worst solu- tion R n+1 .The specific formula is shown in Eq. (28).

Table 3 .
Parameter settings for comparison algorithms.Figure11represents the bar chart of the FT rankings of the 8 algorithms.ERINMRIME possesses the lowest FT index among them in the graph, indicating the stable optimization performance of this algorithm.Figure12displays the convergence curves of ERINMRIME with the other 7 algorithms.The red curve in these convergence plots shows that ERINMRIME consistently remains at the bottom of the graph, exhibiting the highest convergence accuracy and a fast convergence speed.In conclusion, ERINMRIME demonstrates unique competitiveness and significant optimization performance, whether in IEEE CEC 2017 or 2020.

Table 4 .
Avg and Std of ERINMRIME compared with 7 algorithms on IEEE CEC 2017.Significant values are in bold.

Table 5 .
34) I IAE = |I measure − I simulate | WSRT of ERINMRIME compared with 7 algorithms on IEEE CEC 2017.Significant values are in bold.

Table 6 .
Avg and Std of ERINMRIME compared with 7 algorithms on IEEE CEC 2020.Significant values are in bold.

Table 7 .
WSRT of ERINMRIME compared with 7 algorithms on IEEE CEC 2020.Significant values are in bold.

Table 8 .
The boundary of the unknown parameters.

Table 9 .
Parameter extraction results of ERINMRIME on SDM.Significant values are in bold.

Table 10 .
Parameter extraction results of ERINMRIME on DDM.Significant values are in bold.

Table 12
displays the individual RMSE values of ERINMRIME on the PV module model.Additionally, for a thorough assessment, the related comparison curve with other algorithms is shown in Fig.

Table 11 .
Parameter extraction results of ERINMRIME on TDM.Significant values are in bold.

Table 12 .
Parameter extraction results of ERINMRIME on PV module model.Significant values are in bold.
RIME 1.030073 3.33E−06 1.207456 1016.30448.46957 2.43827E−03 + Figure 25.The convergence curve on the PV module model.Vol:.(1234567890)Scientific Reports | (2024) 14:15701 | https://doi.org/10.1038/s41598-024-65292-x EHHO, MLBSA, BSA) at five different irradiation.The specific results are presented in Table14.ERINMRIME exhibits the lowest RMSE in lower irradiation environments and ranks second only to MLBSA in two slightly higher irradiation environments.The results indicate that ERINMRIME exhibits remarkable competitiveness in precisely estimating the parameters of PV models, particularly in low irradiation environments.TableA8(Online Appendix) indicates the specific parameter settings of the PV models for different temperatures.Similarly, ERINMRIME is compared with other algorithms under the same environmental conditions.The comparison findings are presented in Table15.As depicted in the chart, the RMSE of ERINMRIME is the lowest among all the algorithms.ERINMRIME is less affected by temperature variations.ERINMRIME has a certain temperature tolerance and relatively stable measurement effect.Figure

Table 13 .
Time comparison between ERINMRIME and various algorithms.

Table 14 .
RMSE for SM55 various irradiance conditions at 25. Significant values are in bold.

Table 15 .
RMSE for SM55 at various temperatures and 1000 W/m 2 irradiance.Significant values are in bold.

Table 16 .
RMSE at 25 °C for various ST40 irradiation conditions.Significant values are in bold.

Table 17 .
RMSE for ST40 at different temperatures and 1000 W/m 2 irradiance.Significant values are in bold.

Table 18 .
RMSE for KC200GT different irradiance conditions at 25 °C.Significant values are in bold.

Table 19 .
RMSE for KC200GT at different temperatures and 1000 W/m 2 irradiance.Significant values are in bold.